STATISTICAL MODELING OF LARGE WIND PLANT SYSTEM’S GENERATION – A CASE STUDY Dubravko Sabolić Hrvatski operator prijenosnog sustava d.o.o. Croatian transmission System Operator, Ltd. 28 Nov. 2014, Zagreb, Croatia

Goal to obtain simple theoretic distribution functions to model actual statistical distributions of wind generation-related phenomena with sufficient accuracy, so that they can be used for various analyses in either power system operation, or power system economics/policy.

Source of data Bonneville Power Administration (BPA) – federal nonprofit agency based in the Pacific Northwest region of the United States of America. Excel files with historic data of wind generation in 5-minute increments from 2007 on are available at the address: More info on BPA:

BPA-controlled wind plant system More detailed schedule: DATA.pdf

The distributions studied: From the 5-minute readings, recorded from the beginning of 2007 to the end of 2013: – total generation as percentage of total installed capacity; – change in total generation power in 5, 10, 15, 20, 25, 30, 45, and 60 minutes as percentage of total installed capacity; – Limitation of total installed wind plant capacity, when it is determined by regulation demand from wind plants; – duration of intervals with total generated power, expressed as percentage of total installed capacity, lower than certain pre-specified level (I’ll skip this part in the persentation to spare some time. You can read it in the text.) All checked by rigorous tests of goodness-of-fit.

TOTAL GENERATED POWER expressed as percentage of total installed capacity z(x) = e B x A z

TOTAL GENERATED POWER expressed as percentage of total installed capacity Ln(z) = A Lnx + B Figs: 2012

TOTAL GENERATED POWER expressed as percentage of total installed capacity Probability density: y(x) = dz(x)/dx = A e B x A–1 A fat-tailed power-distribution, indicating that extreme values are quite frequent (much more that they would be if distribution was, say, Gaussian).

TOTAL GENERATED POWER expressed as percentage of total installed capacity VAR function: p(r) = e –B/A r 1/A – just an inverse of prob. distr. VaR increases slowly with risk, which is generally a feature most unusual for VaR functions, pointing at unfavorable long-term statistics of wind generation

TEMPORAL CHANGES IN GENERATED POWER in short intervals (intra-hour)

Experimental probability distribution functions, z(x), of short-time changes of wind generation power (2013). Curves are experimental, and the candidate theoretical function is: z(x) = Ln[1 + (e – 1)(1 – e –x/B ) A ]

TEMPORAL CHANGES IN GENERATED POWER in short intervals (intra-hour) Experimental probability density functions, y(x), of short-time changes of wind generation power (2013). Curves are experimental, and the candidate theoretical function is: y(x) = (e – 1)(A/B) (1 – e –x/B ) A–1 /[1 + (e – 1)(1 – e –x/B ) A ].

TEMPORAL CHANGES IN GENERATED POWER in short intervals (intra-hour) Example of goodnes-of-fit (theoretical to experimental): 2013

TEMPORAL CHANGES IN GENERATED POWER in short intervals (intra-hour)

Additional relations between A and B parameters Experimental relations. Ln(A k,year /B k,year ) = – C year Ln(T k /T 5 ) – D year k  {5,10,15,20,25,30,45,60}; C year, D year > 0. Theoretic model. or simply: B k = A k e D (T k /T 5 ) C

TEMPORAL CHANGES IN GENERATED POWER in short intervals (intra-hour) Recalculation of A’s and B’s Experimental relations. Theoretic model. Ln(A k,year /A 60,year ) = –P year Ln 2 (T k /T 60 ) + Q year Ln(T k /T 60 ). or A k  A 60 (T k /T 60 ) Q exp[–P Ln 2 (T k /T 60 )]

TEMPORAL CHANGES IN GENERATED POWER in short intervals (intra-hour) Recalculation of A’s and B’s Experimental relations. Theoretic model. Ln(B k,year /B 60,year ) = R Ln(B k,year /B 60,year ) or B k  B 60 (T k /T 60 ) R

TEMPORAL CHANGES IN GENERATED POWER in short intervals (intra-hour) Recalculation of A’s and B’s What’s the use of these recalculation formulae? How often have you found, say, 15-minute readings of wind plant’s generation? The 60-minute dana are far more available on the Internet. However, they’re not exactly relevant if you wish to study regulation-related issues. So, if you want to go there, you may use recalculation rules to obtain fair approximations to 15-minute readings from 60-minute ones. So, it’s just about being practical. A 15  A Q exp[ P]B 15  B 60 (T 15 /T 60 ) R

Contribution of the wind plant system to the DEMAND FOR REGULATION Let us introduce a concept of REGULATION MULTIPLIER, M reg : How many times can the total installed wind plant power exceede the available fast (secondary) regulation reserve, given certain level of default risk? -Here we shall stick to the wind generation only as a variable parameter, so to grasp at the regulation demand coming from wind generation alone. In reality, one should take into account load variability, too. But then, the wind generation effect would be harder to evaluate separately. -We present here a simplified analysis, assuming the secondary reserve, when needed, is fully cleared by the tertiary one, i.e. fully available at its capacity level. If we kept regulation reserve at x, we would perceive any occurrence of deviation larger than x as a default situation. Therefore, we are interested in: w(x) = 1 – z(x) = 1 – Ln[1 + (e – 1)(1 – e –x/B ) A ]. This function has an inverse: w –1 (r) = – B Ln{1 – [(e 1 – r/100 – 1)/(e – 1)] 1/A }. We formally replaced x by r to denote the risk variable. Finally, we can define the M Reg simply as 100/w –1 (x), or: M Reg = – (100/B) Ln –1 {1 – [(e 1 – r/100 – 1)/(e – 1)] 1/A } This is the VaR function for M reg. A’s and B’s are from the distribution function of temporal changes of generated power.

Contribution of the wind plant system to the DEMAND FOR REGULATION However, due to very complex mathematical operations, it proved a better strategy to first calculate experimental M reg values, 100/[1 – z experimental (r)], and then to use least-squares fit to find appropriate theoretical models. Theoretical model: M Reg = (U/V)r V + W r Obviously, the fit is excelent. Note: In this simplified model it is assumed that the secondary regulation is fully available, that is, fully cleared by the tertiary reserve.

Contribution of the wind plant system to the DEMAND FOR REGULATION Theoretic M Reg functions for 2007, and 2013, and the geometric mean between them.

Thank you for your attention!